Sorption from Solution: A Statistical Thermodynamic Fluctuation Theory

Given an experimental solid/solution sorption isotherm, how can we gain insight into the underlying sorption mechanism on a molecular basis? Classifying sorption isotherms, for both completely and partially miscible solvent/sorbate systems, has been useful, yet the molecular foundation of these classifications remains speculative. Isotherm models, developed predominantly for solid/gas sorption, have been adapted to solid/solution isotherms, yet how their parameters should be interpreted physically has long remained ambiguous. To overcome the inconclusiveness, we establish in this paper a universal theory that can be used for interpreting and modeling solid/solution sorption. This novel theory shares the same theoretical foundation (i.e., the statistical thermodynamic fluctuation theory) not only with solid/gas sorption but also with solvation in liquid solutions and solution nonidealities. The key is the Kirkwood-Buff χ parameter, which quantifies the net self-interaction (i.e., solvent–solvent and sorbate–sorbate interactions minus solvent–sorbate interaction) via the Kirkwood-Buff integral in the same manner as the solvation theory and, unlike the Flory χ, is not limited to the lattice model. We will demonstrate that the Kirkwood-Buff χ is the key not only to isotherm classification but also to generalizing our recent statistical thermodynamic gas (vapor) isotherm, which is capable of fitting most of the solid/solution isotherm types.

(A1) Combining eq A1 with eq 4, we obtain  =  ′ * −  ′  −  ′  −  1 ( 1 * −  1  −  1  ) (A2) Now we carry out the  2 -derivative of eq A2, as where we have emphasized in { } that the mean sorbate numbers have been evaluated in the constant  1 ensembles.Note that we have incorporated the finite distance nature of the interface, which has been denoted by the lowercase for numbers.From now onwards, for simplicity, we do not consider the distribution of sorbate and solvent molecules inside the solid material , hence eq A3 can be simplified as For the evaluation of ( )  , we are under the postulate that the component  does not dissolve into the solution and the species 1 and 2 do not penetrate the solid sorbent.This is in agreement with the common practice that the reference system () is commonly treated like a bulk solution.Consequently, we employ the Gibbs-Duhem equation for the reference system  to evaluate . The results from the Gibbs-Duhem equation can be converted to the { 1  } ensemble due to the ensemble invariance of the mole ratio, which leads to Substituting eq A5 into eq A4 yields the following simplification: This result is compared to another route: a direct differentiation of eq 4 alongside the relationship equivalent to eq A5, which leads to (A7) where Γ 2 (1) is the surface excess (eq 6a).A comparison between eqs A6 and A7 shows that they are two different expressions (in different ensembles) of the same surface excess, Γ 2 (1) .Thus, Γ 2 (1) is invariant under the ensemble transformation.

B. Calculating the gradient of surface excess via ensemble independence.
Our goal is to relate the gradient of surface excess to sorbate number fluctuations.Since Γ 2 is invariant, we carry this out efficiently in the {, ,   ,  where  is the Boltzmann constant.The two number fluctuations in eq B1 can be rewritten in terms of the mole ratio, , and its fluctuation, Since mole ratio fluctuation is invariant under ensemble transformation, the corresponding expression in { 1 } is expressed as Second, we carry out an ensemble transformation from { 1 } to { 1 }.This can be achieved via statistical variable transformation using again the ensemble invariance of the mole ratio, , and its deviation from the mean, 1,2 namely, The Maclaurin expansion of eq B3 yields As discussed in the main text, eq B5 is analogous to the cooperative solubilization theory (e.g., eq 46 of Ref [ 1 ] with the indexes 1 and 2 swapped).Combining eqs A7 and B5, we can write down the fluctuation theory for the gradient of isotherms as ln  2 The solid/solution relationship (eq B6) reduces to the solid/gas counterpart when the solvent is dilute ( 1 * ,  1  → 0 and  1 * ,  1 * → 0), which reduces Γ 2 (1) → 〈 2 * 〉.When the reference phase fluctuation 〈( 2  ) 2 〉 is much smaller than at the interface, 〈( 2 * ) 2 〉, we obtain: when the sorbate excess number is introduced. 3,4Thus, our solid/solution sorption theory contains the solid/gas sorption theory as its special case.
C. The "Kirkwood-Buff " parameter Here we clarify the meaning of eq B6 in terms of the Kirkwood-Buff integrals, defined as where, for the interface, the ensemble is {, ,   ,  1 ,  2 } instead of {, ,   ,  1 ,  2 }.To do so, let us expand the right-hand side of eq B6 and combine it with eq C1, which yields In the main text, we have introduced the Kirkwood-Buff  parameter (eq 8b), through which eq C2 can be rewritten as Using eq C3, eq B6 can be rewritten as ln  2 We will show below that eq C4 can be rewritten into the following form ln  2 where  is the interface/solution sorbate-solvent exchange constant defined as To derive eq C5a, let us start by rewriting eq C4 as ln  2 which is the first term of eq C5a.The second term of eq C6 is Γ 2 (1) via eq 6a, which is the second term of eq C5a.Note that the following relationship between Γ 2 (1) and , Γ 2 (1) = 〈 1 * 〉 2  ( − 1) (C8) will also be useful.

D. The ABC isotherm for solutions.
Here we derive a statistical thermodynamic isotherm equation as an expansion of the interfacial  around  2 = 0. To carry this out, we start with the following: Using eqs C5a and C8, eq D1 can be rewritten as Integrating eq D2 yields where the parameter  has a clear physical interpretation through solvent-surface and sorbate-surface Kirkwood-Buff integrals, defined as where at the dilute limit and  1  is the bulk molar concentration of solvent.
Equation D3 is a general isotherm for adsorption from solution.Here we carry out the activity expansion as where can be interpreted predominantly as the difference of  between interface and solution (see main text).Combining eqs D3 and D5a yields which is the solution-phase generalization of our ABC isotherm for vapor sorption to adsorption form solution.
Our solid/solution parameters are the generalization of our previous theory for solid/gas sorption. 3,4Neglecting  1 leads to a solid/vapor relationship between  2 and the isotherm. 3,4hen the solvent is absent and  * ≫ 1 is dominated by  22 , eq D5b tends to  ≃ ( (D7) which is identical to the vapor-phase theory. 3,4The expression for  involves the three-body correlations that are cumbersome.(Note that  in eq D5a can be expressed in terms of the ensemble averages of numbers and number-ratio by solving eq 8a for .Consequently, the determination of  does not involve any differentiation of .This was already demonstrated in our recent paper on gas (vapor) sorption. 5) In the classification of the isotherms, / plays a key role.To express this parameter in terms of , we must first rewrite eq D4a using in terms of  (eq C5b) as We have used  2 ≃  2  ≃  2  at  2 → 0 at the last step of eq D8.Combining eq D5b and D8, we obtain (D9) which will play a central role in classifying isotherms.
Here we point out that , representing sorbate-water exchange, is indispensable when working with the  differences.It is related also to the solvent-surface and sorbate-surface Kirkwood-Buff integrals via eq D4b, as

E. The ABC isotherm in mole-fraction scale
Let us start with the following equation for  2 , which is analogous to eq D1: where the differentiation on the right-hand side can be linked to the ln  2 derivative as 1,  2 } ensemble (abbreviated as { 1 }) before performing statistical variable transformation back to the {, ,   ,  1 ,  2 } ensemble (abbreviated as { 1 }).First, we differentiate eq A6 in { 1 } with respect to  2 , noting that  1 *